On a class of Hamiltonian polytopes
Let \({\mathcal S}(p,q)\) denote the class of simple 3-polytopal graphs all of whose edges are incident with two p-gons or a p-gon and a q-gon, \(p\neq q\), p,q\(\geq 3\). In this paper it is shown that all graphs of the class \({\mathcal S}(5,q)\) for \(3\leq q\leq 12\), \(q\neq 5\) are Hamiltonian, using the following proposition due to \textit{S. K. Stein}: A simple 3-polytopal graph G is Hamiltonian if and only if its dual \(G^*\) has point- arboricity equal to 2 [Bull. Am. Math. Soc. 76, 805-806 (1970; Zbl 0194.560)]. The theorem in the present paper supplements the known results of \textit{P. R. Goodey} that all graphs of the classes \({\mathcal G}(3,6)\) and \({\mathcal G}(4,6)\) are Hamiltonian, where \({\mathcal G}(p,q)\) denotes the class of 3-connected 3-valent planar graphs, i.e. simple 3-polytopal graphs, all of whose faces are p-gons and q-gons, \(p<q\), \(p\geq 3\) [Isr. J. Math. 22, 52-56 (1975; Zbl 0317.05114); J. Graph Theory 1, 181-185 (1977; Zbl 0379.05037)].
scientific article; zbMATH DE number 4068918